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Area of Semicircle Found From Lines at Right Angles

The diagram shows a semicircle with two lines drawn at right angles. What is the area of the semicircle?

Complete the triangle ABC. From C draw a line to the end of the dimater at D. Triangle ACD is a right angled triangle.
If  
\[\angle BAC =x\]
  then  
\[\angle BCA = 90-x\]
  and  
\[\angle DCB=90-(90-x)=x, \: \angle BDC=90-x\]
.
Triangles ABC and BDC have three identical angles, so are similar triangles.
Then  
\[\frac{AB}{BC}=\frac{BC}{BD} \rightarrow BD=\frac{BC \times BC}{AB}=\frac{16^2}{8}=32\]
 

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